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Hyungju Park

Publications and source records attributed to Hyungju Park.

5 recordsLinked to original sources
arXiv2023

Excluded minors of interval positroids that are paving matroids

We prove that every paving matroid that is an excluded minor of interval positroids can be reduced to one of three fundamental families of excluded minors of interval positroids by relaxing dependent hyperplanes. Using this result, we classify all non-positroid excluded minors of interval positroids that are paving matroids. Additionally, we provide a criterion that characterizes all excluded minors of interval positroids that are paving positroids.

math.CO↗
arXiv2023

Almost generalized uniform matroids and excluded minors

We establish that matroids characterized by the Tutte polynomial $\sum_{i,j\ge 0}t_{i,j}x^iy^j$ with coefficients $t_{i,j}$ vanishing for $(i,j)\ge (k,l)$ precisely coincide with $(k,l)$-uniform matroids. This characterization implies that almost $(k,l)$-uniform matroids are exactly matroids with $t_{k,l}\le 1$ and $t_{i,j}=0$ if $(i,j)>(k,l)$. We also characterize excluded minors of almost $(k,l)$-uniform matroids in terms of Tutte polynomial coefficients. Finally, we construct an infinite family of excluded minors of almost $(k,l)$-uniform matroids which extend previously known cases of almost uniform and almost paving matroids.

math.CO↗
arXiv2017

Grothendieck-Plücker images of Hilbert schemes are degenerate

We study the decompositions of Hilbert schemes induced by the Schubert cell decomposition of the Grassmannian variety and show that Hilbert schemes admit a stratification into locally closed subschemes along which the generic initial ideals remain the same. We give two applications: First, we give a completely geometric proofs of the existence of the generic initial ideals and of their Borel fixed properties. Secondly, we prove that when a Hilbert scheme of nonconstant Hilbert polynomial is embedded by the Grothendieck-Plücker embedding of a high enough degree, it must be degenerate.

math.AG↗
arXiv1998

Algorithms for Determining Birationality of Parametrization of Affine Curves

Let $k$ be an arbitrary field, and C be a curve in A^n defined parametrically by x_1=f_1(t),...,x_n=f_n(t), where f_1,...,f_n\in k[t]. A necessary and sufficient condition for the two function fields k(t) and k(f_1,...,f_n) to be same is developed in terms of zero-dimensionality of a derived ideal in the bivariate polynomial ring k[s,t]. Since zero-dimensionality of such an ideal can be readily determined by a Groebner basis computation, this gives an algorithm that determines if the parametrization ψ=(f_1,...,f_n): A --> C is a birational equivalence. We also develop an algorithm that determines if k[t] and k[f_1,...,f_n] are same, by which we get an algorithm that determines if the parametrization ψ=(f_1,...,f_n): A --> C is an isomorphism. We include some computational examples showing the application of these algorithms.

math.AG↗